county courthouse vs county seat

county courthouse and county seat both are nouns.

Word	Noun	Adjective	Verb	Adverb
county courthouse	Yes	No	No	No
county seat	Yes	No	No	No

As nouns, county courthouse and county seat are synonyms defined as:

county courthouse and county seat: the town or city that is the seat of government for a county

county courthouse (noun)	county seat (noun)
the town or city that is the seat of government for a county	the town or city that is the seat of government for a county